The Spectral Action and the Standard Model

Transcription

1 Ma148b Spring 2016 Topics in Mathematical Physics

2 References A.H. Chamseddine, A. Connes, M. Marcolli, Gravity and the Standard Model with Neutrino Mixing, Adv. Theor. Math. Phys., Vol.11 (2007)

3 The spectral action functional Ali Chamseddine, Alain Connes, The spectral action principle, Comm. Math. Phys. 186 (1997), no. 3, A good action functional for noncommutative geometries Tr(f (D/Λ)) D Dirac, Λ mass scale, f > 0 even smooth function (cutoff approx) Simple dimension spectrum expansion for Λ Tr(f (D/Λ)) f k Λ k D k + f (0) ζ D (0) + o(1), k with f k = 0 f (v) v k 1 dv momenta of f where DimSp(A, H, D) = poles of ζ b,d (s) = Tr(b D s )

4 Asymptotic expansion of the spectral action Tr(e t ) a α t α (t 0) and the ζ function ζ D (s) = Tr( s/2 ) Non-zero term a α with α < 0 pole of ζ D at 2α with Res s= 2α ζ D (s) = 2 a α Γ( α) No log t terms regularity at 0 for ζ D with ζ D (0) = a 0

5 Asymptotic expansion of the spectral action Tr(e t ) a α t α (t 0) and the ζ function ζ D (s) = Tr( s/2 ) Non-zero term a α with α < 0 pole of ζ D at 2α with Res s= 2α ζ D (s) = 2 a α Γ( α) No log t terms regularity at 0 for ζ D with ζ D (0) = a 0

6 Get first statement from D s = s/2 = 1 Γ ( ) s 2 with 1 0 tα+s/2 1 dt = (α + s/2) 1. Second statement from 1 Γ ( s 2) s 2 0 as s 0 contrib to ζ D (0) from pole part at s = 0 of 1 given by a 0 0 ts/2 1 2 dt = a 0 s 0 Tr(e t ) t s/2 1 dt e t t s/2 1 dt

7 Spectral action with fermionic terms S = Tr(f (D A /Λ)) J ξ, D A ξ, ξ H + cl, D A = Dirac with unimodular inner fluctuations, J = real structure, H + cl = classical spinors, Grassmann variables Fermionic terms 1 2 J ξ, D A ξ antisymmetric bilinear form A( ξ) on H + cl = {ξ H cl γξ = ξ} nonzero on Grassmann variables Euclidean functional integral Pfaffian Pf (A) = e 1 2 A( ξ) D[ ξ] avoids Fermion doubling problem of previous models based on symmetric ξ, D A ξ for NC space with KO-dim=0

8 Grassmann variables Anticommuting variables with basic integration rule ξ dξ = 1 An antisymmetric bilinear form A(ξ 1, ξ 2 ): if ordinary commuting variables A(ξ, ξ) = 0 but not on Grassmann variables Example: 2-dim case A(ξ, ξ) = a(ξ 1 ξ 2 ξ 2 ξ 1), if ξ 1 and ξ 2 anticommute, with integration rule as above e 1 2 A(ξ,ξ) D[ξ] = e aξ 1ξ 2 dξ 1 dξ 2 = a Pfaffian as functional integral: antisymmetric quadratic form Pf (A) = e 1 2 A(ξ,ξ) D[ξ] Method to treat Majorana fermions in the Euclidean setting

9 Fermionic part of SM Lagrangian Explicit computation of 1 2 J ξ, D A ξ gives part of SM Larangian with L Hf = coupling of Higgs to fermions L gf = coupling of gauge bosons to fermions L f = fermion terms

10 Bosonic part of the spectral action S = 1 π 2 (48 f 4 Λ 4 f 2 Λ 2 c + f 0 g 4 d) d 4 x + 96 f 2 Λ 2 f 0 c 24π 2 R g d 4 x f π 2 ( 11 6 R R 3 C µνρσ C µνρσ ) g d 4 x + ( 2 a f 2 Λ 2 + e f 0 ) π 2 ϕ 2 g d 4 x + f 0a 2 π 2 D µ ϕ 2 g d 4 x f 0 a 12 π 2 R ϕ 2 g d 4 x + f 0b 2 π 2 ϕ 4 g d 4 x f π 2 (g3 2 Gµν i G µνi + g2 2 Fµν α F µνα g 1 2 B µν B µν ) g d 4 x,

11 Parameters: f 0, f 2, f 4 free parameters, f 0 = f (0) and, for k > 0, f k = 0 f (v)v k 1 dv. a, b, c, d, e functions of Yukawa parameters of νmsm a = Tr(Y ν Y ν + Y e Y e + 3(Y u Y u + Y d Y d)) b = Tr((Y ν Y ν ) 2 + (Y e Y e ) 2 + 3(Y u Y u ) 2 + 3(Y d Y d) 2 ) c = Tr(MM ) d = Tr((MM ) 2 ) e = Tr(MM Y ν Y ν ).

12 Gilkey s theorem using DA 2 = E Differential operator P = (g µν I µ ν + A µ µ + B) with A, B bundle endomorphisms, m = dim M Tr e tp n 0 t n m 2 P = E and E ; µ µ := µ µ E M a n (x, P) dv(x) µ = µ + ω µ, ω µ = 1 2 g µν(a ν + Γ ν id) Seeley-DeWitt coefficients E = B g µν ( µ ω ν + ω µ ω ν Γ ρ µν ω ρ) Ω µν = µ ω ν ν ω µ + [ω µ, ω ν] a 0 (x, P) = (4π) m/2 Tr(id) a 2 (x, P) = (4π) m/2 Tr ( R 6 id + E) a 4 (x, P) = (4π) m/ Tr( 12R ;µ µ + 5R 2 2R µν R µν + 2R µνρσ R µνρσ 60 R E E E ; µ µ + 30 Ω µν Ω µν )

13 Normalization and coefficients Rescale Higgs field H = a f0 π ϕ to normalize kinetic term 1 2 D µh 2 g d 4 x Normalize Yang-Mills terms 1 4 G µνg i µνi F µνf α µνα B µνb µν Normalized form: 1 S = 2κ 2 R g g d 4 x + γ 0 d 4 x 0 + α 0 C µνρσ C µνρσ g d 4 x + τ 0 R R g d 4 x + 1 DH 2 g d 4 x µ 2 0 H 2 g d 4 x 2 ξ 0 R H 2 g d 4 x + λ 0 H 4 g d 4 x + 1 (Gµν i G µνi + Fµν α F µνα + B µν B µν ) g d 4 x 4 where R R = 1 4 ɛµνρσ ɛ αβγδ R αβ µνr γδ ρσ integrates to the Euler characteristic χ(m) and C µνρσ Weyl curvature

14 Coefficients 1 = 96f 2Λ 2 f 0 c 2κ π 2 γ 0 = 1 π 2 (48f 4Λ 4 f 2 Λ 2 c + f 0 4 d) α 0 = 3f 0 10π 2 τ 0 = 11f 0 60π 2 µ 2 0 = 2f 2Λ 2 e a f 0 λ 0 = π2 b 2f 0 a 2 ξ 0 = 1 12 Energy scale: Unification ( GeV) g 2 f 0 2π 2 = 1 4 Preferred energy scale, unification of coupling constants

15 1-loop RGE equations for νmsn variable t = log(λ/m Z ) Coupling constants: t x i (t) = β xi (x(t)) β 1 = 41 96π 2 g 3 1, β 2 = 19 96π 2 g 3 2, β 3 = 7 16π 2 g 3 3 at 1-loop decoupled from other equations (Notation: g 2 1 = 5 3 g 2 1 ) Yukawa parameters: 16π 2 β Yu = Y u ( 3 2 Y u Y u 3 2 Y d Y d + a g g 2 2 8g 2 3 ) 16π 2 β Yd = Y d ( 3 2 Y d Y d 3 2 Y u Y u + a 1 4 g g 2 2 8g 2 3 ) 16π 2 β Yν = Y ν ( 3 2 Y ν Y ν 3 2 Y e Y e + a 9 20 g g 2 2 ) 16π 2 β Ye = Y e ( 3 2 Y e Y e 3 2 Y ν Y ν + a 9 4 g g 2 2 )

16 Majorana mass terms: Higgs self coupling: 16π 2 β M = Y ν Y ν M + M(Y ν Y ν ) T 16π 2 β λ = 6λ 2 3λ(3g g 2 1 ) + 3g (g g 2 2 ) 2 + 4λa 8b MSM approximation: top quark Yukawa parameter dominant term: in λ running neglect all terms except coupling constants g i and Yukawa parameter of top quark y t β λ = 1 16π 2 ( 24λ λy 2 9λ(g g 2 1 ) 6y g g g 2 2 g 2 1 where Yukawa parameter for the top quark runs by β y = 1 ( 9 16π 2 2 y 3 8g3 2 y 9 4 g 2 2 y 17 ) 12 g 1 2 y )

17 Renormalization group flow The coefficients a, b, c, d, e (depend on Yukawa parameters) run with the RGE flow Initial conditions at unification energy: compatibility with physics at low energies RGE in the MSM case Running of coupling constants at one loop: α i = g 2 i /(4π) β gi = (4π) 2 b i g 3 i, with b i = ( 41 6, 19 6, 7), α 1 1 (Λ) = α 1 1 (M Z ) 41 12π log α 1 2 (Λ) = α 1 2 (M Z ) π log α 1 3 (Λ) = α 1 3 (M Z ) π log M Z GeV mass of Z 0 boson Λ M Z Λ M Z Λ M Z

18 At one loop RGE for coupling constants decouples from Yukawa parameters (not at 2 loops!) Couplings 5 3 Α Α2 Α log 10 Μ GeV Well known triangle problem: with known low energy values constants don t meet at unification g 2 3 = g 2 2 = 5g 2 1 /3

19 Geometry point of view At one loop coupling constants decouple from Yukawa parameters Solving for coupling constants, RGE flow defines a vector field on moduli space C 3 C 1 of Dirac operators on the finite NC space F Subvarieties invariant under flow are relations between the SM parameters that hold at all energies At two loops or higher, RGE flow on a rank three vector bundle (fiber = coupling constants) over the moduli space C 3 C 1 Geometric problem: studying the flow and the geometry of invariant subvarieties on the moduli space

20 Constraints at unification The geometry of the model imposes conditions at unification energy: specific to this NCG model λ parameter constraint Higgs vacuum constraint af0 λ(λ unif ) = π2 2f 0 b(λ unif ) a(λ unif ) 2 π = 2M W g See-saw mechanism and c constraint 2f 2 Λ 2 unif c(λ unif ) 6f 2Λ 2 unif f 0 f 0 Mass relation at unification (mν 2 + me 2 + 3mu 2 + 3md 2 ) Λ=Λ unif = 8MW 2 Λ=Λ unif generations Need to have compatibility with low energy behavior

21 Mass relation at unification Y 2 (S) = 4g 2 Y 2 = σ (y σ ν ) 2 + (y σ e ) (y σ u ) (y σ d )2 (k ( 3) ) σκ = g 2M mσ u δσ κ (k ( 3) ) σκ = g 2M mµ d C σµδµc ρ ρκ (k ( 1) ) σκ = g 2M mσ ν δσ κ (k ( 1) ) σκ = g 2M mµ e U lep σµδµu ρ lep ρκ δ j i = Kronecker delta, then constraint: Tr(k ( 1) k ( 1) + k ( 1) k ( 1) + 3(k ( 3) k ( 3) + k ( 3) k ( 3))) = 2 g 2 mass matrices satisfy (mν σ ) 2 + (me σ ) (mu σ ) (md σ )2 = 8 M 2 σ

22 See-saw mechanism: D = D(Y ) Dirac 0 M ν M R 0 M ν M R 0 0 M ν 0 0 M ν 0 on subspace (ν R, ν L, ν R, ν L ): largest eigenvalue of M R Λ unification scale. Take M R = x k R in flat space, Higgs vacuum v small (w/resp to unif scale) u Tr(f (D A /Λ)) = 0 u = x 2 Dirac mass M ν Fermi energy v x 2 = 2 f 2 Λ 2 Tr(k R k R) f 0 Tr((k R k R) 2 ) 1 2 (±m R ± m 2 R + 4 v 2 ) two eigenvalues ±m R and two ± v 2 m R Compare with estimates (m R ) GeV, (m R ) GeV, (m R ) GeV

23 Low energy limit: compatibilities and predictions Running of top Yukawa coupling (dominant term): v 2 (y σ ) = (m σ ), dy t dt = 1 [ 9 16π 2 2 y t 3 ( ] a g1 2 + b g2 2 + c g3 2 ) yt, (a, b, c) = ( 17 12, 9 4, 8) value of top quark mass agrees with known (1.04 times if neglect other Yukawa couplings)

24 Top quark running using mass relation at unification yt yt log 10 Μ GeV correction to MSM flow by yν σ for τ neutrino (allowed to be comparably large by see-saw) lowers value

25 Higgs mass prediction using RGE for MSM Higgs scattering parameter: f 0 2 π 2 b ϕ 4 g d 4 x = π2 2 f 0 b a 2 H 4 g d 4 x relation at unification ( λ is H 4 coupling) λ(λ) = g 2 3 b a 2 Running of Higgs scattering parameter: dλ dt = λγ + 1 8π 2 (12λ2 + B) γ = 1 16π 2 (12y 2 t 9g 2 2 3g 2 1 ) B = 3 16 (3g g 2 1 g g 4 1 ) 3y 4 t

26 Higgs estimate (in MSM approximation for RGE flow) m 2 H = 8λ M2 g 2, m H = 2λ 2M g x log 10 Μ GeV 0.28 Λ log Μ MZ x 0.2 λ(m Z ) and Higgs mass 170 GeV (w/correction from see-saw 168 GeV)... Problem: wrong Higgs mass! too heavy